Biomacromolecules 2004, 5, 780-785
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Modeling of Mechanical Properties and Structural Design of Spider Web Frank K. Ko* and Jovan Jovicic† Department of Materials Science and Engineering, Drexel University, Philadelphia, Pennsylvania 19104 Received December 7, 2003; Revised Manuscript Received March 1, 2004
With a unique combination of strength and toughness among materials, spider silk is the model for engineering materials. This paper presents the stress-strain behavior of Nephila claVipes spider silk under tension, transverse compression, and torsional deformation obtained by a battery of micro testing equipment. The experimental results showed significantly higher toughness than the state-of-the-art fibers in tension and in transverse compression. Higher shear modulus was also observed for the spider silk comparing to other liquid crystalline fibers such as aramid fibers. On the basis of the experimental results finite element analysis is used to simulate static and dynamic properties of spider web and to explore the role of both material properties and architectural design in its structural integrity and mechanical performance. Introduction In the world of natural fibers, spider silk has long been recognized as the wonder fiber for its unique combination of high strength and rupture elongation. An earlier study indicated spider silk has strength as high as 1.75 GPa at a breaking elongation of over 26%.1,2 With toughness more than three times that of aramid and other industrial fibers, spider silk continues to attract the attention of fiber scientists and hobbyists alike.3-12 Apart from the remarkable material properties of spider silk, spider webs are natural example of a special class of pre-stressed systems called tensegrity (tensional integrity) structures.13-14 These structures represent a unique blend of geometry and mechanics, resulting in highly efficient structures due to the optimal distribution of structural mass. The geometry plays a major role in defining the existence as well as the stiffness of a tensegrity structure. The pretension acts little to change the stiffness. However, it plays an important role in delaying the onset of slack strings and thus prevents the nonlinearities due to slackening of the strings. The tensegrity systems can be considered as space structures. Their lightness places them in the same class as cable and membrane systems. The self-stressing nature, which provides their rigidity, provides spider webs the mechanism for efficient and economic means of balancing the stresses induced. An understanding of the interaction of material properties and structural geometry may shed light on our ability to design the next generation of ultra-lightweight, large area space structures (Figure 1). Draglines of N. claVipes and A. aurentia spider silks are among the strongest spider silks that we know. The strength of the dragline of N. claVipes obtained by forcible silking was reported to be about 8 g/denier, and the strength of the * To whom correspondence should be addressed. E-mail: fko@ coe.drexel.edu. † Present address: Department of Mechanical Engineering, Widener University, Chester, PA 19013.
dragline of A. aurentia was approximately 12 g/denier.1 Considering the remarkable mechano-chemical properties of spider silk and fueled by the recent progress in biotechnology, there is a revival of interest in using spider silk as a model for the engineering of high energy absorption fibers.15 To obtain realistic blueprints for the regenerated or synthetic spider silk, there is a need for precise characterization of the engineering properties of spider silk. However, because of the fineness of spider silk, less than 4 µm, the characterization of the mechanical properties of spider silks are traditionally limited to tensile mode. Little is known about the response of spider silks to other modes of deformation in the transverse direction and in torsion. This article offers a summary of an initial attempt to characterize the stressstrain behavior of from N. claVipes spiders under simple tension, transverse compression, and torsional deformation. This was made possible by using an ultra sensitive micromeasurement fiber testing system developed by Kawabata.16 From these experimental data, the engineering propertiess tensile modulus, transverse compressive modulus, and shear modulussof the spider silk were determined. These engineering properties of spider silk provided a basis for the structural analysis of spider webs. To guide the analysis of experimental results and the assessment of the design of the spider web systems, a computational methodology was developed using finite element analysis. It is hoped that this study will lead us to an improved understanding of the relative significance of the fiber material properties and the structural parameter of a spider web. Engineering Properties One of the outstanding characteristics of spider silk is its fineness. For example the dragline is between 3 and 4 microns in diameter. The cribellate silk was found to be as fine as 0.03 µm in diameter. The drag-line silks have a circular fiber cross-section and a density of 1.25 g/cc. The smallness of the spider silk makes it difficult to handle the
10.1021/bm0345099 CCC: $27.50 © 2004 American Chemical Society Published on Web 04/08/2004
Modeling of Spider Web Properties and Design
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Figure 1. Initial FE model and boundary conditions.
Figure 3. Tensile property of N. clavipes spider single fiber.
Figure 2. Tensile stress-strain curves of A. aurentia spider major ampulate gland silk and other polyamide fibers.
fibers and thus challenging in preparing specimens for the characterization of their engineering properties under various modes of loading. In this paper, the engineering properties of a single spider silk were characterized using a series of micro-testers. Figure 2 shows the stress-strain curve of the A. aurentia spider silk assuming a sigmoidal shape similar to that of an elastomer. Comparing to other aramid fibers, it demonstrates a superior balance of strength and elongation at 1.75 GPa (15.8 g/den) and 36%, respectively. This “rubberlike” stress-strain curve is characterized by three distinct regions: region I (0-5%) is characterized by a high initial modulus of 34 GPa; region II (5-21%) shows a pseudo yield point at 5% before strain hardening to a maximum modulus of 22 GPa at 22% elongation; and region III (21-36%) exhibits a gradual reduction of modulus until reaching failure strength of 1.75 GPa at 36% elongation. An examination of the area under the stress-strain curves shows a toughness level of 2.8 g/denier. This is much higher than the toughness of the aramid fiber (0.26 g/denier) and nylon-6 fiber (0.9 g/denier).
Tensile Properties. The silk from a N. claVipes spider obtained from the US Army Natick RD&E Laboratories was tested using the micro-tensile tester in Professor Kawabata’s laboratory. The silks were collected at Natick according to the procedure detailed by Cunniff et al.11 The spider silk was tested by simple elongation at a strain rate of 100% per minute using a gage length of 1.25 cm. Additionally, transverse compression and torsional properties of the N. claVipes spider silk were also evaluated. Ten replications of the N. claVipes spider drag-line silk were made to generate the average tensile stress-strain curve shown in Figure 3 wherein a sigmoidal shape stress-strain curve similar to that of the A. aurentia spider is shown. With an average initial modulus of 12.71 GPa, the failure stress of the fiber is 0.85 GPa at 20% breaking elongation. Similar to that of the A. aurentia spider, a yield point is detected at about 5% strain. These results show that the maximum properties of spider silk may differ from one species to the other with the A. aurentia spider making stronger silk than the N. claVipes spider. This could be affected by the silking process as a result of the degree of drawing. However, the sigmoidal “rubber-like” shape of the stress-strain curves appears to be a common feature characterizing spider drag-line silk. In comparison with other textile fibers (as shown in Figure 4), similar to that exhibited by the A. aurentia spider silk,
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Figure 4. Tensile stress-strain behavior of N. clavipes spider major ampulate gland silk compared to other textile fibers.
Ko and Jovicic
Figure 6. Torsion tester for single fiber.
Figure 7. Torsional stress-strain behavior of N. clavipes spider silk. Table 1. Engineering Properties of Polyamide Fibers
Figure 5. Compressive stress-strain behavior of N. clavipes spider silk.
the N. claVipes silk exhibits the best balance of strength and toughness. Transverse Compression Properties. The compression tests of the spider silks in the transverse direction were carried out by placing a single fiber between a flat and mirror-finished steel plate and a mirror finished 0.2 mm square compression plane. Because of the fineness of the spider fiber, a combination of sensitive instrumentation and mechanistic analysis is required in order to ensure accurate measurement of the compressive stress-stain properties. A description of the compression tester has been detailed by Kawabata.16 The N. claVipes spider silk fibers were subjected to transverse cyclic loading at a compressive speed of 0.3 cm/s under ambient and wet conditions. The compressive modulus of the fiber tested in ambient condition was 0.58 GPa and the fiber experienced a high degree of permanent deformation (∼20%). As shown in Figure 5, the ability of spider silk to deform transverse compression is higher than all of the other textile fibers, indicating a high level of anisotropy and transverse ductility, offering superior ability to absorb energy
material
EL (GPa)
A. aurentia spider silk N. clavipes spider silk B. mori silk merino wool nylon 6 filament Kevlar 29
34.00 12.71 9.90 3.50 2.71 79.80
ET (GPa)
GL (GPa)
0.579
2.38 3.81 1.31 0.52 2.17
0.93 1.01 2.59
EL/ET
E L /GL
21.95
5.34 4.93 2.67 5.21 36.77
3.76 2.68 30.81
under deformation in the transverse direction such as in the crossover between silk fibers. Torsional Properties. Through torsional testing, the shear modulus of a fiber can be determined. The torsional behavior of the N. claVipes spider silk was characterized with an ultrasensitive Kawabata torsional tester. As shown in Figure 6, a single fiber having both ends reinforced by a paper backing using ceramic adhesives is hung on a top hook connected to a highly sensitive torque detector supported by two torque wires made of 0.2 mm piano wire. The bottom end is connected to a bar, and both ends of the bar are inserted into slits of a servo-driven cylindrical tube. The full scale of the torque meter is 0.0025 gf cm/10 V. A high level of torsional resistance is observed for the spider silk. The shear rigidity, as determined from the torque-deformation diagram shown in Figure 7, is 2.38 GPa, which is higher than all of the other textile fibers including Kevlar 29. This appears to be consistent with the intended use of the drag-line as a lifeline for the spider (as in a mountain climbing rope) which requires a high level of torsional stability.
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Modeling of Spider Web Properties and Design
Figure 8. Stress-strain curve using the ductile mode. Figure 10. Stress-strain properties of a web segment under the impact for various fibers.
Table 2. Engineering Properties of the Compared Materials
material spider silk viscid dragline Kevlar 29 spectra PBO
density [kg/m3] 1098 1450 970 1540
tensile modulus EL (GPa) 0.003 34 45 101 270
Poisson’s ratio ν
strength σ (GPa)
strain %
0.49
0.58 1.75 2.8 3.0 5.8
150 26 3.6 3.4 3.5
0.34 0.4 0.35
Figure 11. Strain energy as a function of impact time for different web materials in the elastic domain.
Figure 9. Strain energy as a function of web density.
Numerical Modeling of a Spider Web With the engineering properties experimentally characterized, one can carry out structural analysis of the spider web and examine the unique combination of engineering properties of spider silk in comparison with other manmade fibers. Table 1 provides a summary of the engineering properties of the spider silks in comparison with other natural and synthetic polyamide fibers. To address the influence of different design parameters on spider’s web structural integrity and complement experimental efforts, a finite element method (FEM) is employed. More specifically the ABAQUS-EXPLICIT finite element code (Hibbitt Inc., Rhode Island, 2001) is used to simulate the static and dynamic properties of the spider web and to explore the role of material properties and architectural design in its structural integrity and mechanical performance. As a first approximation, as shown in Figure 1, the spider web is modeled as an elasto-plastic 3D truss structure (3,865 elements and 2,031 nodes), with a fiber diameter of 3.57 µm and fully constrained foundation lines.
Simulation of the impact of a bug on a web with 5 N pretension is performed using the initial velocity of 1 m/s. The bug-impactor is modeled as a spherical rigid body with a mass of 0.5 g. In these numerical models, radial web components are made of dragline silk, whereas web concentric circles (a capture silk) are assumed to be made of viscid silk material. The material properties used in simulations are listed in Table 2. The mechanical properties of the spider silk obtained from tensile tests are compared to those of the man-made fibers. Initially, the isotropic elastic-plastic constitutive model for manmade fibers is assumed. The fibers were modeled using damage-Von Mises plasticity with a specific plastic strain failure. The ductile failure model is based on a damage-Von Mises plasticity theory with isotropic hardening. The damage manifests itself in two forms: degradation of the yield stress with damage and damaged elasticity. Figure 8 illustrates the deviatoric stress/strain behavior of the material model. The solid curve represents the actual stress, whereas the dashed curve represents undamaged behavior, that is, when only elasto-plasticity is considered. Until the initial yield stress, σ0, is reached, the fiber behaves as an elastic material. Plastic strain than occurs according to the conventional Von Mises plasticity theory. If the strain continues to increase, damage will increase from zero, when the plastic strain is less than or equal to the offset plastic strain, pl 0 , to a value of one when the
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material points in the element fail, the element loses its ability to resist any further load, and hence, it is removed from the mesh. Accordingly, a parametric study is performed using the FE modeling design tool developed herein by considering (1) different materials (Spider silk, Kevlar, PBO, Spectra) and (2) various densities of the web as characterized by the aspect ratio of its circular constituents (ranging from low to medium to high). Modeling Results Figure 12. Strain energy as a function of impact time for different web materials in the plastic range of deformation.
plastic strain reaches the plastic failure strain, pl f . At that time, the corresponding total strain is f. The damage (D) is calculated from the plastic strain as D)
pl - pl pl - pl
(1)
where pl is the current plastic strain experienced by the fiber. The material’s elastic response is based on damaged elasticity. The damaged elastic modulus is given as ED ) (1 - D)E
(2)
When the plastic strain at a material point reaches the plastic failure strain, pl f , the material point has failed. If all of the
For an impact velocity of 1 m/s and a duration of 0.01s, the elastic response of the web structure is observed. The total strain energy is evaluated as a function of time for different web geometry (density of the web) as shown in Figure 9 and for different web materials listed in Table 2 (Figure 10). Because of its relatively high stiffness, compared those of to the other fibers, PBO has the highest elastic strain energy absorption as shown in Figure 11. On the other hand, as shown in Figure 12, the spider silk has the highest strain energy in the plastic region compared to those of the other fibers listed in Table 2. These preliminary results show that the excellent combination of strength and toughness of a spider silk offers great potential and superior dynamic characteristics compared to other materials that have higher tensile strength than that of
Figure 13. Principal stress distribution during simulated impact on spider silk web.
Figure 14. Principal stress distribution during simulated impact on PBO web.
Modeling of Spider Web Properties and Design
the spider silk. The superior strain energy is especially prominent in the plastic region. This effect is illustrated using spider silk and PBO fiber webs as examples by subjecting them to impact loading involving deformations up to the plastic range for duration of 0.1 s by a 2 g bug impacting the web at a velocity of 1 m/s (Figure 12). The finite element analysis predicted that the spider web would be able to stop the impacting bug without failure (Figure 13). In the same time, for the same boundary and loading condition, simulations showed that the web made of PBO material would be perforated (Figure 14). Conclusions In summary, with the engineering properties characterized and the engineering design tool created, we have established the basis for asking further questions on the structural design of spider webs. Preliminary finite element simulations of impact properties of spider webs showed that the unique combination of both high toughness and stiffness proved to be superior compared to other current state-of-the-art manmade materials. Strategic tailoring of the material properties for the frame (E ) 10 GPa; f ) 17.4-35%) and viscid silk (E ) 3 MPa; f ) 150%) should be further explored. Apart from structural applications, the weblike structures could be employed to predict mechanical behavior and accurately detect and locate damage by transmitting and sensing propagating plastic waves. The numerical model developed herein offers a possibility to evaluate the effect of fiber material properties, fiber diameter, as well as the effect of web geometry (e.g., concentric circular vs spiral web architecture) on the structural performance of the web and the sensing capabilities of the tensegrity structures.
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References and Notes (1) Zemlin, J. C. A Study of the Mechanical BehaVior of Spider Silks; U. S. Army Natick Technical Report 69-29-CM, 1968. (2) Ko, F. Nonlinear Viscoelasticity of Aramid Fibers. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1977. (3) Witt, P. N.; Reed, C. F.; Peakall, D. B. A Spider’s Web: Problems in Regulatory Biology; Springer-Verlag: New York, l968. (4) Friedrich, V. L., Jr.; Langer, R. M. Fine Structure of Cribellate Spider SiIk. Am. Zool. 1969, 9, 91. (5) Peakall, D. B. Synthesis of Silk, Mechanism and Location Am. Zool. 1969, 9, 71. (6) Lucas, F.; Shaw, J. T. B.; Smith, S. G. Comparative Studies of Fibroins: I. The Amino Acid Composition of Various Fibroins and Its Significance in Relation to Their Crystal Structure and Taxonomy. J. Mol. Biol. 1960, 2, 339. (7) Marples, B. J. The Spinnerets and Epiandrous Glands of Spiders. J. Linnean Soc. (Zool.) 1967, 46, 209. (8) Wilson, R. S. The Structure of the Dragline Control Valves in the Garden Spiders. Q. J. Micr. Sci. 1962, 104, 549. (9) Wilson, R. S. The Control of Drag-line Spinning in the Garden Spiders. Q. J. Micr. Sci. 1962, 104, 557. (10) Wilson, R. S. Control of Drag-line Spinning in Certain Spiders. Am. Zool. 1969, 9, 103. (11) Cunniff, P. M.; Fossey, S. A.; Auerbach, M. A.; Song, J. W. Mechanical Properties of Majos Ampulate Gland Silk Fibers Extracted from Nephila clapvipes Spiders. In Silk Polymers: Materials Science and Biotechnology; Kaplan, D., Adams, W. W., Famler, B., Viney, C., Eds.; ACS Symposium Series 544; American Chemical Society: Washington, DC, 1994; Chapter 21. (12) Vollrath, F.; Knight, D. Liquid Crystalline Spinning of Spider Silk. Nature 2001, 410, 541-548. (13) Motro, R. Tensegrity Systems: The State of the Art. Int. J. Space Struct. 1992, 7 (2), 1992. (14) Gosline, J. M.; DeMont, M. E.; Deuny, M. W. EndeaVor 1986, 10, 37-43. (15) Kaplan, D., Adams, W. W., Famler, B., Viney, C., Eds.; Silk Polymers: Materials Science and Biotechnology; ACS Symposium Series 544; American Chemical Society: Washington, DC, 1994. (16) Kawabata, S. Micromeasurement of Mechanical Properties of Single Fibers. In Modern Textile Characterization Methods of High Performance Fibers; Raheel, M., Ed.; Marcel Dekker: New York, 1996; pp 311-328.
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